Miller, Kierste – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert
1
This
printout
should
have
17
questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
beFore answering.
The due time is Central
time.
YES, homework 2 is due BEFORE
homework 1
001
(part 1 oF 1) 10 points
Determine
A
so that the curve
y
= 7
x
+ 17
can be written in parametric Form as
x
(
t
) =
t

3
,
y
(
t
) =
At

4
.
1.
A
= 9
2.
A
=

7
3.
A
= 8
4.
A
=

9
5.
A
=

8
6.
A
= 7
correct
Explanation:
We have to eliminate
t
From the parametric
equations For
x
and
y
. Now From the equation
For
x
it Follows that
t
=
x
+ 3. Thus
y
= 7
x
+ 17 =
A
(
x
+ 3)

4
.
Consequently
A
= 7
.
keywords:
Cartesian equation, parametric
equations, eliminate parameter
002
(part 1 oF 1) 10 points
Determine a Cartesian equation For the
curve given in parametric Form by
x
(
t
) = 4
e
2
t
,
y
(
t
) = 3
e

t
.
1.
x
y
2
= 48
2.
xy
2
= 48
3.
xy
2
= 36
correct
4.
x
y
2
= 36
5.
x
2
y
= 12
6.
x
2
y
= 12
Explanation:
We have to eliminate the parameter
t
From
the equations For
x
and
y
.
Now From the
equation For
x
it Follows that
e
t
=
‡
x
4
·
1
/
2
,
±rom which in turn it Follows that
y
= 3
‡
4
x
·
1
/
2
.
Consequently,
x
2
y
= 36
.
keywords:
Cartesian equation, parametric
equations, eliminate parameter
003
(part 1 oF 1) 10 points
Describe the motion oF a particle with posi
tion
P
(
x, y
) when
x
= 5 sin
t,
y
= 3 cos
t
as
t
varies in the interval 0
≤
t
≤
2
π
.
1.
Moves once clockwise along the ellipse
x
2
25
+
y
2
9
= 1
,
starting and ending at (0
,
3).
correct
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentMiller, Kierste – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert
2
2.
Moves once clockwise along the circle
(5
x
)
2
+ (3
y
)
2
= 1
,
starting and ending at (0
,
3).
3.
Moves once counterclockwise along the
ellipse
x
2
25
+
y
2
9
= 1
,
starting and ending at (0
,
3).
4.
Moves along the line
x
5
+
y
3
= 1
,
starting at (5
,
0) and ending at (0
,
3).
5.
Moves along the line
x
5
+
y
3
= 1
,
starting at (0
,
3) and ending at (5
,
0).
6.
Moves once counterclockwise along the
circle
(5
x
)
2
+ (3
y
)
2
= 1
,
starting and ending at (0
,
3).
Explanation:
Since
cos
2
t
+ sin
2
t
= 1
for all
t
, the particle travels along the curve
given in Cartesian form by
x
2
25
+
y
2
9
= 1 ;
this is an ellipse centered at the origin.
At
t
= 0, the particle is at (5 sin 0
,
3 cos 0),
i.e.
,
at the point (0
,
3) on the ellipse.
Now as
t
increases from
t
= 0 to
t
=
π/
2,
x
(
t
) increases
from
x
= 0 to
x
= 5, while
y
(
t
) decreases from
y
= 3 to
y
= 0 ; in particular, the particle
moves from a point on the positive
y
axis to
a point on the positive
x
axis, so it is moving
clockwise
.
In the same way, we see that as
t
increases
from
π/
2 to
π
, the particle moves to a point
on the negative
y
axis, then to a point on the
negative
x
axis as
t
increases from
π
to 3
π/
2,
until Fnally it returns to its starting point on
the positive
y
axis as
t
increases from 3
π/
2 to
2
π
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Gilbert
 Sin, Parametric equation, Conic section

Click to edit the document details